scholarly journals Newton-Kantorovich Iterative Regularization for Nonlinear Ill-Posed Equations Involving Accretive Operators

2005 ◽  
Vol 57 (2) ◽  
pp. 323-330 ◽  
Author(s):  
Buong Nguyen ◽  
Quang Vu
Keyword(s):  
Accretive Operators ◽  
Iterative Regularization ◽  
Ill Posed

scholarly journals Heating source localization in a reduced time

2016 ◽  
Vol 26 (3) ◽  
pp. 623-640 ◽  
Author(s):  
Sara Beddiaf ◽  
Laurent Autrique ◽  
Laetitia Perez ◽  
Jean-Claude Jolly
Keyword(s):  
Heat Conduction ◽  
Source Localization ◽  
Conjugate Gradient ◽  
Regularization Method ◽  
Three Dimensional ◽  
Gradient Algorithm ◽  
Iterative Regularization ◽  
Heating Source ◽  
Ill Posed ◽  
Reduced Time

Abstract Inverse three-dimensional heat conduction problems devoted to heating source localization are ill posed. Identification can be performed using an iterative regularization method based on the conjugate gradient algorithm. Such a method is usually implemented off-line, taking into account observations (temperature measurements, for example). However, in a practical context, if the source has to be located as fast as possible (e.g., for diagnosis), the observation horizon has to be reduced. To this end, several configurations are detailed and effects of noisy observations are investigated.

scholarly journals ITERATED LAVRENTIEV REGULARIZATION FOR NONLINEAR ILL-POSED PROBLEMS

2009 ◽  
Vol 51 (2) ◽  
pp. 191-217 ◽  
Author(s):  
P. MAHALE ◽  
M. T. NAIR
Keyword(s):  
Nonlinear Operator ◽  
Optimal Error Estimate ◽  
Positive Real ◽  
Iterative Regularization ◽  
Optimal Error ◽  
Lavrentiev Regularization ◽  
Inexact Data ◽  
Ill Posed ◽  
Funct Anal ◽  
General Source

AbstractWe consider an iterated form of Lavrentiev regularization, using a null sequence (αk) of positive real numbers to obtain a stable approximate solution for ill-posed nonlinear equations of the form F(x)=y, where F:D(F)⊆X→X is a nonlinear operator and X is a Hilbert space. Recently, Bakushinsky and Smirnova [“Iterative regularization and generalized discrepancy principle for monotone operator equations”, Numer. Funct. Anal. Optim.28 (2007) 13–25] considered an a posteriori strategy to find a stopping index kδ corresponding to inexact data yδ with $\|y-y^\d \|\leq \d $ resulting in the convergence of the method as δ→0. However, they provided no error estimates. We consider an alternate strategy to find a stopping index which not only leads to the convergence of the method, but also provides an order optimal error estimate under a general source condition. Moreover, the condition that we impose on (αk) is weaker than that considered by Bakushinsky and Smirnova.

Iterative Regularization Methods for Nonlinear Ill-Posed Problems

2008 ◽  
Author(s):  
Barbara Kaltenbacher ◽  
Andreas Neubauer ◽  
Otmar Scherzer
Keyword(s):  
Regularization Methods ◽  
Iterative Regularization ◽  
Ill Posed

scholarly journals A LEPSKIJ-TYPE STOPPING-RULE FOR SIMPLIFIED ITERATIVELY REGULARIZED GAUSS-NEWTON METHOD

2017 ◽  
Author(s):  
Agah D. Garnadi
Keyword(s):  
Newton Method ◽  
Hilbert Spaces ◽  
Potential Problem ◽  
Initial Approximation ◽  
Stopping Rule ◽  
A Posteriori ◽  
Iterative Regularization ◽  
Ill Posed ◽  
Derivatives Of ◽  
Unknown Solution

Iterative regularization methods for nonlinear ill-posed equations of the form $ F(a)= y$, where $ F: D(F) \subset X \to Y$ is an operator between Hilbert spaces $ X $ and $ Y$, usually involve calculation of the Fr\'{e}chet derivatives of $ F$ at each iterate and at the unknown solution $ a^\sharp$. A modified form of the generalized Gauss-Newton method which requires the Fr\'{e}chet derivative of $F$ only at an initial approximation $ a_0$ of the solution $ a^\sharp$ as studied by Mahale and Nair \cite{MaNa:2k9}. This work studied an {\it a posteriori} stopping rule of Lepskij-type of the method. A numerical experiment from inverse source potential problem is demonstrated.

scholarly journals On fractional asymptotical regularization of linear ill-posed problems in hilbert spaces

2019 ◽  
Vol 22 (3) ◽  
pp. 699-721 ◽  
Author(s):  
Ye Zhang ◽  
Bernd Hofmann
Keyword(s):  
Fractional Order ◽  
Hilbert Spaces ◽  
Regularization Method ◽  
Numerical Realization ◽  
Regularization Methods ◽  
Iterative Regularization ◽  
Operator Equations ◽  
Ill Posed ◽  
One Step ◽  
The One

Abstract In this paper, we study a fractional-order variant of the asymptotical regularization method, called Fractional Asymptotical Regularization (FAR), for solving linear ill-posed operator equations in a Hilbert space setting. We assign the method to the general linear regularization schema and prove that under certain smoothness assumptions, FAR with fractional order in the range (1, 2) yields an acceleration with respect to comparable order optimal regularization methods. Based on the one-step Adams-Moulton method, a novel iterative regularization scheme is developed for the numerical realization of FAR. Two numerical examples are given to show the accuracy and the acceleration effect of FAR.

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